HYBRID MODELING OF COMMUNICATION NETWORKS USING MODELICA
Daniel F¨arnqvist Katrin Strandemar Karl Henrik Johansson Jo˜ao Pedro Hespanha
Department of Signals, Sensors & Systems Royal Institute of Technology, Stockholm, Sweden
kallej@s3.kth.se
Department of Electrical & Computer Engineering University of California, Santa Barbara, USA
hespanha@ece.ucsb.edu
Abstract: Modeling and simulation of communication networks using the modeling language Modelica are discussed. Congestion control in packet-switched networks, such as the Internet, is today mainly analyzed through time-consuming simulations of individual packets. We show, by developing a model library based on a recent hybrid systems model, that Modelica provides an efficient platform for the analysis of communication networks. As an example, a comparison between the two congestion control protocols is presented.
Keywords: Congestion control; Packet-switched networks; TCP; Internet; Modelica;
Computer simulation; Hybrid systems
1. INTRODUCTION
The objective with research on control of networks is often to improve traffic throughput and to bet- ter accommodate different service demands. Com- munication networks experience major problems due to traffic congestion. Today’s congestion control is in most networks implemented as end-to-end proto- cols (Jacobson, 1988; Walrand and Varaiya, 2000; Pe- terson and Davie, 2000). The protocols have proved to form the basis of a remarkably robust and scal- able system, though the understanding of the basic principles of these complex systems are far from satisfactory (Paxson and Floyd, 1995; Paxson and Floyd, 1997; Paganini et al., 2001).
The intention of this paper is to describe initial work on modeling packet-switched communication network using Modelica (Modelica, n.d.; Mattsson et al., 1998), which is an object-oriented language for modeling physical systems. Standard modeling and
simulation environments targeted at network research include the discrete-event simulator ns-2 (The Net- work Simulator ns-2, n.d.). Since ns-2 directly imple- ments the Internet protocols and simulates individual packets, it provides on one hand accurate simulation results but on the other hand a rather slow simulation speed. The result of this is that ns-2 is mainly for studying relatively small networks over a short time scale. The other extreme is to use flow models, i.e., to approximate packet transmission with a continuous flow and basically neglect the network protocols. A hybrid systems model, which is based on the average rates but takes packet drops and rate adjustments due to congestion control into account, was recently pro- posed in the literature (Hespanha et al., 2001). The motivation for this model is to capture the network behavior on a time scale in between packet models and flow models. Studies have shown that the hybrid model is able to model many important network phe- nomena (Hespanha et al., 2001; Bohacek et al., 2001).
In this paper, we show that the hybrid model is suitable for Modelica. Moreover, we show that the model can be efficiently simulated using Dymola (Elmqvist et al., n.d.), which is a commercial Modelica simulation environment.
The outline of the paper is as follows. In Section 2, a brief introduction to congestion control in communi- cation networks is presented. The hybrid model is de- scribed in Section 3 and its Modelica implementation is discussed in Section 4. An example, where two TCP versions are compared for a small wireless network, is given in Section 5. The paper ends in Section 6 with a brief discussion on ongoing work. An early version of this paper was presented as (F¨arnqvist et al., 2002)
2. COMMUNICATION NETWORKS A packet-switched network can be described by a graph, see Figure 1. The nodes represent the routers, which direct the packets from sender to receiver, and the edges correspond to wired or wireless links. The bandwidths of the links are limited, so each router has a buffer where packets are stored if more packets are entering the router than are leaving. In this way, it is possible to deal with minor traffic congestion in the network. If too many packets enter a router in a short amount of time, however, packets will be dropped due to that the router buffer has finite size. The way this congestion problem is handled by the senders on the Internet is through a control mechanism denoted transmission control protocol (TCP).
In TCP the receiver sends acknowledgments back to the sender, when packets have arrived. In order to efficiently use the network resources, the TCP sender adjusts its sending rate according to a control variable called the window size w. The TCP sender sends w number of packets and waits for their acknowledg- ments to return. Hence, w corresponds to the number of unacknowledged packets the sender may have in the network. When the sender has received acknowl- edgments for all w packets, the window size is in- creased by one: w : w 1. If a packet is dropped (so that no acknowledgment for that packet is re- ceived by the sender), w is decreased by a factor two.
Hence, TCP uses additive increase and multiplicative decrease (AIMD) to regulate the congestion window size based on explicit acknowledgments and implicit negative acknowledgments. Although, the AIMD con- trol strategy has proved to be efficient, robust and re- markably scalable for the Internet, it is believed that it might be too abrupt for emerging applications such as streaming of audio and video. Moreover, using regular TCP over wireless links has shown to be inefficient, see further discussion in Section 5.
Fig. 1. Communication network. Packets from a sender to a receiver are transfered over wired or wireless links connected by routers. Since the links and the routers have limited capacity, con- gestion control strategies are needed.
3. HYBRID MODEL
In the hybrid model for communication networks pro- posed by Hespanha et al. (Hespanha et al., 2001; Bo- hacek et al., 2001), the network dynamics and the TCP dynamics are modeled as hybrid systems, e.g., (Witsenhausen, 1966; Brockett, 1993; van der Schaft and Schumacher, 2000; Johansson et al., 2002). A hybrid system is a mathematical model for a dynamic system that has both continuous-time and discrete- event dynamics. The traffic dynamics of a packet- switched communication network is suitable to model as a hybrid system, where the packet flows are approx- imated by continuous dynamics while packet drops and control protocols are modeled as discrete changes.
The motivation for introducing a hybrid model is that it captures the behavior of the system on a time scale in between conventional packet models and flow models.
Thanks to the model abstraction of packets, the simu- lation time is in principle not affected by the number of packets transmitted or the queue sizes. Still, the hybrid model shows enough detail to accurately model transmission protocols and packet drops.
3.1 Network Dynamics
Packet transmission rates and queue lengths are mod- eled as continuous-time real-valued variables in the hybrid model. The arrival rate of packets at a router is denoted s and the departure rate is denoted r. The number of packets stored in a router queue is denoted q. The dynamics of the queue is depending on if the queue is full (q qmax) or not (0 q qmax). For each router and flow, we introduce the hybrid system with two discrete states shown in Figure 2. In this model, subscript f refers to flow f , the variables q and s are defined as q ∑fqfand s ∑fsf, and the bandwidth of the outgoing link is equal to a constant B. Note that when the queue is full, a drop will be generated as soon as the variable z is equal to the predefined packet size L. Which flow f of the incoming flows that will
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? L z = :=0
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Fig. 2. Hybrid queue model. Each router is modeled by a number of two-state hybrid systems, each representing the queue dynamics for an individ- ual flow f . As long as the queue is not full, the evolution of the queue size qf is simply ˙qf sf
rf.
lose a packet is determined by the distribution of the flows in the queue.
3.2 TCP Dynamics
For TCP, the sending rate r and the window size w are conveniently modeled as continuous variables, while the various modes of TCP (such as slow-start, conges- tion avoidance, and fast recovery) represent discrete states. The hybrid TCP model consists of four discrete states as shown in Figure 3. Here the round-trip time RT Tf for flow f is the time between sending a packet and receiving the corresponding acknowledgment. It is given by the sum of the physical propagation time and the queueing times. The hybrid TCP model uses RT Tf to update the window size wf and to derive the sending rate rf. These variables also depend on the discrete state of the TCP. Most of the time TCP is in the congestion avoidance state. Here wf grows approximately linearly ( ˙wf 1 RT Tf), since RT Tf is almost constant. When a drop occurs, wf is reduced by a factor two.
3.3 Simplest Example
Let us illustrate TCP with the simplest example, namely, a single queue and a single flow. This system consists of one hybrid queue model of Figure 2 and one hybrid TCP model of Figure 3. Figure 4 shows a simulation of the queue size q (dashed) and the window size w (solid) for a typical TCP session. When the queue becomes full (q qmax 57), a packet is dropped and w is reduced by a factor two. The evolution of the window size has a sawtooth shape (thanks to the AIMD control strategy) characteristic for TCP flows. Note the exponential increase of w in
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Fig. 3. Hybrid TCP model. Each TCP flow is mod- eled by a hybrid system with four discrete states.
During a regular transmission session, the system is most of the time in the congestion avoidance state. In that state the window size grows approx- imately linearly: ˙wf 1 RT Tf.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 10 20 30 40 50 60 70 80 90 100
time
packets
q w
Fig. 4. Illustration of congestion avoidance for a net- work with a single router and a single flow. When the queue q (dashed) exceeds qmax 57, a packet is dropped. TCP reduces accordingly the window w (solid) by a factor two. Note the characteristic shape of w with intervals of approximately linear growth and periodic jumps.
the beginning of the session. The reason for this is that TCP starts in the slow-start state, which (in spite of its name) gives a rapid increase of the sending rate (and accordingly, the window size) in the initial phase of a session.
4. MODELICA IMPLEMENTATION Modelica is an object-oriented modeling language de- signed mainly for large heterogeneous physical sys- tems. It is suitable for the hybrid network model de- scribed in previous section. For example, the continu-
rf
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Routing
1 TCP 2
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2 TCP sf
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Fig. 5. Composition of hybrid communication net- work model. The network dynamics and the TCP dynamics are separated.
ous dynamics of the TCP model in Figure 3 is imple- mented through the following code:
if CongAvoid then der(w)=1/RTT;
r=w*L/RTT;
elseif FastRecov then der(w)=0;
r=w*L/RTT;
elseif TimeOut then der(w)=0;
r=0;
elseif SlowStart then der(w)=log(2)*w/RTT;
r=w*L/RTT;
end if;
Discrete events, such as the packet drops in the net- work model, can be implemented in the Modelica language as the following example:
when drop then reinit(w,w/2);
end when;
A communication network library was developed in Modelica. The library contains standard build- ing blocks for network simulation, such as TCP senders, routing tables, and queues. Figure 5 shows the schematic layout of our communication network model. Note that the network dynamics and the TCP dynamics are separated. The only information shared between the two submodels are the sending rates rf, the round-trip times RT Tf, and the drop events. The modular structure allows an easy testing of for in- stance different TCP controllers applied to the same network topology. Note that only the queue and the TCP components contain dynamical equations. The router component implement the network topology.
The complexity of the model depends on the number of routers and the number of TCP flows. Note that each router–flow pair requires a hybrid queue model of Figure 2.
Appropriate handling of the switching between dis- crete states is important for accurate and efficient sim- ulation of hybrid systems (Johansson et al., 1999a; Jo- hansson et al., 1999b). Implementations in Simulink
Fig. 6. Example of a communication network imple- mented in Modelica.
showed some problems in this respect. Our implemen- tation in Modelica and simulation in Dymola works well. Note that the simulation time is in general not depending on the queue sizes or sending rates, but in- stead depends on the number of discrete events gener- ated by packet drops. The simulation time grows con- siderably when the number of discrete events becomes large. In ns-2, where individual packets are simulated, the simulation time depends on the size of the packet flows, the number of flows, and the queue sizes.
5. WIRELESS EXAMPLE
An example of a communication network imple- mented in Modelica is shown in Figure 6, where two senders and two receivers are connected to the network. Let us simulate this simple network. The two flows are sharing the same link capacity. If the sum of the flows at some time instance is larger than the bandwidth of the link, packets will be queued.
Sender 1 sends Flow f1using a version of TCP called TCP Westwood (TCPW) (Mascolo et al., 2001) and Sender 2 sends Flow f2using TCP SACK (Hespanha et al., 2001). For a wireless link transmission losses are more likely than for a wired link. For the network in Figure 6, there can be packet losses either due to that the router queue is full or due to that the wireless transmission loses packets. We model the wireless link as having a good and a bad state. In the good state 0.1% of the transmitted packets are lost, while in the bad state 10% of the packets are lost. The link stays in each state a random amount of time, which is exponentially distributed. TCPW was designed taking wireless links into account, while TCP SACK was designed for wired links. This is illustrated next.
0 5 10 15 20 25 0
10 20 30 40 50 60 70
time
w
TCPW TCP SACK
Fig. 7. Simulation of the system in Figure 6. The windows size for TCP SACK is considerably reduced when the wireless link is losing a lot of packets (at about t 5 and t 15), while TCPW is able to cope with the losses very well.
0 5 10 15 20 25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
2x 107
time
Throughput
TCPW TCP SACK
Fig. 8. TCPW gives a larger throughput than TCP SACK. The difference is emphasized when there are a large number of packet losses due to the wireless transmission.
Figure 7 shows the window sizes for a simulation of the network in Figure 6. The solid line corresponds to TCPW and the dashed line to TCP SACK. Note the time intervals at about t 5 and t 15, when the link is in the bad state. The packet losses due to the bad transmission result in a sudden decrease of the window size for TCP SACK, while TCPW are able to compensate for the packet losses of the wireless link.
The window size for TCPW is in general larger than for TCP SACK. This gives a larger throughput for the connection using TCPW, as is shown in Figure 8.
Figure 9 shows the throughput when two TCPW’s are sharing the same wireless link (upper plots) and when two TCP SACK’s are sharing the same link (lower plots). From the simulations we see that the major advantage of TCPW is when the link is in the bad state.
When the link is in the good state, the performance of both TCP implementations are roughly equal.
0 5 10 15 20 25
0 2 4 6 8 10 12 14 16x 106
time
Throughput
TCPW 1 TCPW 2 TCP SACK 1 TCP SACK 2
Fig. 9. Throughput for two TCP Westwood senders (upper plots) and two TCP SACK senders (lower plots).
Since TCP (SACK) was developed for wired net- works, where packet losses arise mainly due to buffer overflow, there is a problem using it over wireless links. The reason is that the window size is reduced by a factor two every time a drop occurs, regardless if the drop is due to congestion or to transmission loss.
Hence, the sending rate r will be reduced even if it is not needed. TCPW has another way of updating the window size when a drop occurs. The window size is set to a value based on an estimate of the available bandwidth and the current round-trip time RT T . Since RT T is highly dependent on the queue sizes, so that RT T is small only if all corresponding queues are small, a small RT T implies that a detected drop must be due to a wireless loss. The aim of TCPW is hence to utilize the available bandwidth more efficiently.
6. DISCUSSION
Heterogeneous networks with both wired and wireless links are important for many emerging applications.
Ongoing work includes the comparison of various TCP variants, such as Reno, SACK, and Westwood, but also the proposal of improved congestion control mechanisms. An idea for router-controlled congestion avoidance is presented in (Auvert et al., 2002), where so called network flaps are introduced to smoothing out bursty traffic flows.
The modeling environment that we have developed based on Modelica provides a flexible modular plat- form for simulating various networks and protocols.
Note, however, that the hybrid model is not only suit- able for efficient simulation, but also provides a math- ematical framework suitable for analysis using tools from hybrid systems theory.
Acknowledgment
The authors want to thank H˚akan Hjalmarsson and Gunnar Karlsson for helpful discussions. The work by Karl Henrik Johansson was supported by the Swedish Research Council.
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